Trigonometry Examples

$- \frac{\sqrt{1 - \cos (6 x)}}{1 + \cos (6 x)}$

Step 1

Interchange the variables.

$x = - \frac{\sqrt{1 - \cos (6 y)}}{1 + \cos (6 y)}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \frac{\sqrt{1 - \cos (6 y)}}{1 + \cos (6 y)} = x$ .

$- \frac{\sqrt{1 - \cos (6 y)}}{1 + \cos (6 y)} = x$

Step 2.2

Cross multiply.

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Step 2.2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

$x \cdot (1 + \cos (6 y)) = - \sqrt{1 - \cos (6 y)}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify $x \cdot (1 + \cos (6 y))$ .

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Step 2.2.2.1.1

Apply the distributive property.

$x \cdot 1 + x \cos (6 y) = - \sqrt{1 - \cos (6 y)}$

Step 2.2.2.1.2

Multiply $x$ by $1$ .

$x + x \cos (6 y) = - \sqrt{1 - \cos (6 y)}$

Step 2.3

Rewrite the equation as $- \sqrt{1 - \cos (6 y)} = x + x \cos (6 y)$ .

$- \sqrt{1 - \cos (6 y)} = x + x \cos (6 y)$

Step 2.4

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{1 - \cos (6 y)})}^{2} = {(x + x \cos (6 y))}^{2}$

Step 2.5

Simplify each side of the equation.

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - \cos (6 y)}$ as ${(1 - \cos (6 y))}^{\frac{1}{2}}$ .

${(- {(1 - \cos (6 y))}^{\frac{1}{2}})}^{2} = {(x + x \cos (6 y))}^{2}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Simplify ${(- {(1 - \cos (6 y))}^{\frac{1}{2}})}^{2}$ .

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Step 2.5.2.1.1

Apply the product rule to $- {(1 - \cos (6 y))}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(1 - \cos (6 y))}^{\frac{1}{2}})}^{2} = {(x + x \cos (6 y))}^{2}$

Step 2.5.2.1.2

Raise $- 1$ to the power of $2$ .

$1 {({(1 - \cos (6 y))}^{\frac{1}{2}})}^{2} = {(x + x \cos (6 y))}^{2}$

Step 2.5.2.1.3

Multiply ${({(1 - \cos (6 y))}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(1 - \cos (6 y))}^{\frac{1}{2}})}^{2} = {(x + x \cos (6 y))}^{2}$

Step 2.5.2.1.4

Multiply the exponents in ${({(1 - \cos (6 y))}^{\frac{1}{2}})}^{2}$ .

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Step 2.5.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(1 - \cos (6 y))}^{\frac{1}{2} \cdot 2} = {(x + x \cos (6 y))}^{2}$

Step 2.5.2.1.4.2

Cancel the common factor of $2$ .

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Step 2.5.2.1.4.2.1

Cancel the common factor.

${(1 - \cos (6 y))}^{\frac{1}{2} \cdot 2} = {(x + x \cos (6 y))}^{2}$

Step 2.5.2.1.4.2.2

Rewrite the expression.

${(1 - \cos (6 y))}^{1} = {(x + x \cos (6 y))}^{2}$

Step 2.5.2.1.5

Simplify.

$1 - \cos (6 y) = {(x + x \cos (6 y))}^{2}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Simplify ${(x + x \cos (6 y))}^{2}$ .

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Step 2.5.3.1.1

Rewrite ${(x + x \cos (6 y))}^{2}$ as $(x + x \cos (6 y)) (x + x \cos (6 y))$ .

$1 - \cos (6 y) = (x + x \cos (6 y)) (x + x \cos (6 y))$

Step 2.5.3.1.2

Expand $(x + x \cos (6 y)) (x + x \cos (6 y))$ using the FOIL Method.

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Step 2.5.3.1.2.1

Apply the distributive property.

$1 - \cos (6 y) = x (x + x \cos (6 y)) + x \cos (6 y) (x + x \cos (6 y))$

Step 2.5.3.1.2.2

Apply the distributive property.

$1 - \cos (6 y) = x \cdot x + x (x \cos (6 y)) + x \cos (6 y) (x + x \cos (6 y))$

Step 2.5.3.1.2.3

Apply the distributive property.

$1 - \cos (6 y) = x \cdot x + x (x \cos (6 y)) + x \cos (6 y) x + x \cos (6 y) (x \cos (6 y))$

Step 2.5.3.1.3

Simplify and combine like terms.

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Step 2.5.3.1.3.1

Simplify each term.

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Step 2.5.3.1.3.1.1

Multiply $x$ by $x$ .

$1 - \cos (6 y) = x^{2} + x (x \cos (6 y)) + x \cos (6 y) x + x \cos (6 y) (x \cos (6 y))$

Step 2.5.3.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.3.1.3.1.2.1

Move $x$ .

$1 - \cos (6 y) = x^{2} + x \cdot x \cos (6 y) + x \cos (6 y) x + x \cos (6 y) (x \cos (6 y))$

Step 2.5.3.1.3.1.2.2

Multiply $x$ by $x$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x \cos (6 y) x + x \cos (6 y) (x \cos (6 y))$

Step 2.5.3.1.3.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.3.1.3.1.3.1

Move $x$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x \cdot x \cos (6 y) + x \cos (6 y) (x \cos (6 y))$

Step 2.5.3.1.3.1.3.2

Multiply $x$ by $x$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x^{2} \cos (6 y) + x \cos (6 y) (x \cos (6 y))$

Step 2.5.3.1.3.1.4

Multiply $x$ by $x$ by adding the exponents.

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Step 2.5.3.1.3.1.4.1

Move $x$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x^{2} \cos (6 y) + x \cdot x \cos (6 y) \cos (6 y)$

Step 2.5.3.1.3.1.4.2

Multiply $x$ by $x$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x^{2} \cos (6 y) + x^{2} \cos (6 y) \cos (6 y)$

Step 2.5.3.1.3.1.5

Multiply $x^{2} \cos (6 y) \cos (6 y)$ .

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Step 2.5.3.1.3.1.5.1

Raise $\cos (6 y)$ to the power of $1$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x^{2} \cos (6 y) + x^{2} (\cos^{1} (6 y) \cos (6 y))$

Step 2.5.3.1.3.1.5.2

Raise $\cos (6 y)$ to the power of $1$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x^{2} \cos (6 y) + x^{2} (\cos^{1} (6 y) \cos^{1} (6 y))$

Step 2.5.3.1.3.1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x^{2} \cos (6 y) + x^{2} {\cos (6 y)}^{1 + 1}$

Step 2.5.3.1.3.1.5.4

Add $1$ and $1$ .

$1 - \cos (6 y) = x^{2} + x^{2} \cos (6 y) + x^{2} \cos (6 y) + x^{2} \cos^{2} (6 y)$

Step 2.5.3.1.3.2

Add $x^{2} \cos (6 y)$ and $x^{2} \cos (6 y)$ .

$1 - \cos (6 y) = x^{2} + 2 x^{2} \cos (6 y) + x^{2} \cos^{2} (6 y)$

Step 2.6

Solve for $y$ .

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Step 2.6.1

Substitute $u$ for $\cos (6 y)$ .

$1 - (u) = x^{2} + 2 x^{2} (u) + x^{2} {(u)}^{2}$

Step 2.6.2

Since $u$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 2 x^{2} u + x^{2} u^{2} = 1 - u$

Step 2.6.3

Add $u$ to both sides of the equation.

$x^{2} + 2 x^{2} u + x^{2} u^{2} + u = 1$

Step 2.6.4

Subtract $1$ from both sides of the equation.

$x^{2} + 2 x^{2} u + x^{2} u^{2} + u - 1 = 0$

Step 2.6.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.6.6

Substitute the values $a = x^{2}$ , $b = 2 x^{2} + 1$ , and $c = x^{2} - 1$ into the quadratic formula and solve for $u$ .

$\frac{- (2 x^{2} + 1) \pm \sqrt{{(2 x^{2} + 1)}^{2} - 4 \cdot (x^{2} \cdot (x^{2} - 1))}}{2 x^{2}}$

Step 2.6.7

Simplify the numerator.

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Step 2.6.7.1

Apply the distributive property.

$u = \frac{- (2 x^{2}) - 1 \cdot 1 \pm \sqrt{{(2 x^{2} + 1)}^{2} - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.2

Multiply $2$ by $- 1$ .

$u = \frac{- 2 x^{2} - 1 \cdot 1 \pm \sqrt{{(2 x^{2} + 1)}^{2} - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.3

Multiply $- 1$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{{(2 x^{2} + 1)}^{2} - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.4

Rewrite ${(2 x^{2} + 1)}^{2}$ as $(2 x^{2} + 1) (2 x^{2} + 1)$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{(2 x^{2} + 1) (2 x^{2} + 1) - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.5

Expand $(2 x^{2} + 1) (2 x^{2} + 1)$ using the FOIL Method.

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Step 2.6.7.5.1

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 x^{2} (2 x^{2} + 1) + 1 (2 x^{2} + 1) - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.5.2

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2} + 1) - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.5.3

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6

Simplify and combine like terms.

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Step 2.6.7.6.1

Simplify each term.

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Step 2.6.7.6.1.1

Rewrite using the commutative property of multiplication.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 x^{2} x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.6.7.6.1.2.1

Move $x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 (x^{2} x^{2})) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 x^{2 + 2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.1.2.3

Add $2$ and $2$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 x^{4}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.1.3

Multiply $2$ by $2$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.1.4

Multiply $2$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.1.5

Multiply $2 x^{2}$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} + 2 x^{2} + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.1.6

Multiply $1$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} + 2 x^{2} + 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.6.2

Add $2 x^{2}$ and $2 x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.7.7

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{2} x^{2} - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.7.8

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.6.7.8.1

Move $x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 (x^{2} x^{2}) - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.7.8.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{2 + 2} - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.7.8.3

Add $2$ and $2$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{4} - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.7.9

Multiply $- 1$ by $- 4$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{4} + 4 x^{2}}}{2 x^{2}}$

Step 2.6.7.10

Subtract $4 x^{4}$ from $4 x^{4}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{2} + 1 + 0 + 4 x^{2}}}{2 x^{2}}$

Step 2.6.7.11

Add $4 x^{2}$ and $0$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{2} + 1 + 4 x^{2}}}{2 x^{2}}$

Step 2.6.7.12

Add $4 x^{2}$ and $4 x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.6.8.1

Change the $\pm$ to $+$ .

$u = \frac{- 2 x^{2} - 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.8.2

Factor $- 1$ out of $- 2 x^{2}$ .

$u = \frac{- (2 x^{2}) - 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.8.3

Rewrite $- 1$ as $- 1 (1)$ .

$u = \frac{- (2 x^{2}) - 1 \cdot 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.8.4

Factor $- 1$ out of $- (2 x^{2}) - 1 (1)$ .

$u = \frac{- (2 x^{2} + 1) + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.8.5

Factor $- 1$ out of $\sqrt{8 x^{2} + 1}$ .

$u = \frac{- (2 x^{2} + 1) - 1 (- \sqrt{8 x^{2} + 1})}{2 x^{2}}$

Step 2.6.8.6

Factor $- 1$ out of $- (2 x^{2} + 1) - 1 (- \sqrt{8 x^{2} + 1})$ .

$u = \frac{- (2 x^{2} + 1 - \sqrt{8 x^{2} + 1})}{2 x^{2}}$

Step 2.6.8.7

Rewrite $- (2 x^{2} + 1 - \sqrt{8 x^{2} + 1})$ as $- 1 (2 x^{2} + 1 - \sqrt{8 x^{2} + 1})$ .

$u = \frac{- 1 (2 x^{2} + 1 - \sqrt{8 x^{2} + 1})}{2 x^{2}}$

Step 2.6.8.8

Move the negative in front of the fraction.

$u = - \frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.6.9.1

Simplify the numerator.

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Step 2.6.9.1.1

Apply the distributive property.

$u = \frac{- (2 x^{2}) - 1 \cdot 1 \pm \sqrt{{(2 x^{2} + 1)}^{2} - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.2

Multiply $2$ by $- 1$ .

$u = \frac{- 2 x^{2} - 1 \cdot 1 \pm \sqrt{{(2 x^{2} + 1)}^{2} - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.3

Multiply $- 1$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{{(2 x^{2} + 1)}^{2} - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.4

Rewrite ${(2 x^{2} + 1)}^{2}$ as $(2 x^{2} + 1) (2 x^{2} + 1)$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{(2 x^{2} + 1) (2 x^{2} + 1) - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.5

Expand $(2 x^{2} + 1) (2 x^{2} + 1)$ using the FOIL Method.

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Step 2.6.9.1.5.1

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 x^{2} (2 x^{2} + 1) + 1 (2 x^{2} + 1) - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.5.2

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2} + 1) - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.5.3

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 x^{2} (2 x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6

Simplify and combine like terms.

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Step 2.6.9.1.6.1

Simplify each term.

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Step 2.6.9.1.6.1.1

Rewrite using the commutative property of multiplication.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 x^{2} x^{2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.6.9.1.6.1.2.1

Move $x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 (x^{2} x^{2})) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 x^{2 + 2}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.1.2.3

Add $2$ and $2$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{2 \cdot (2 x^{4}) + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.1.3

Multiply $2$ by $2$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} \cdot 1 + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.1.4

Multiply $2$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} + 1 (2 x^{2}) + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.1.5

Multiply $2 x^{2}$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} + 2 x^{2} + 1 \cdot 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.1.6

Multiply $1$ by $1$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 2 x^{2} + 2 x^{2} + 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.6.2

Add $2 x^{2}$ and $2 x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 \cdot x^{2} \cdot (x^{2} - 1)}}{2 x^{2}}$

Step 2.6.9.1.7

Apply the distributive property.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{2} x^{2} - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.9.1.8

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 2.6.9.1.8.1

Move $x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 (x^{2} x^{2}) - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.9.1.8.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{2 + 2} - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.9.1.8.3

Add $2$ and $2$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{4} - 4 x^{2} \cdot - 1}}{2 x^{2}}$

Step 2.6.9.1.9

Multiply $- 1$ by $- 4$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{4} + 4 x^{2} + 1 - 4 x^{4} + 4 x^{2}}}{2 x^{2}}$

Step 2.6.9.1.10

Subtract $4 x^{4}$ from $4 x^{4}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{2} + 1 + 0 + 4 x^{2}}}{2 x^{2}}$

Step 2.6.9.1.11

Add $4 x^{2}$ and $0$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{4 x^{2} + 1 + 4 x^{2}}}{2 x^{2}}$

Step 2.6.9.1.12

Add $4 x^{2}$ and $4 x^{2}$ .

$u = \frac{- 2 x^{2} - 1 \pm \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.9.2

Change the $\pm$ to $-$ .

$u = \frac{- 2 x^{2} - 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.9.3

Factor $- 1$ out of $- 2 x^{2}$ .

$u = \frac{- (2 x^{2}) - 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.9.4

Rewrite $- 1$ as $- 1 (1)$ .

$u = \frac{- (2 x^{2}) - 1 \cdot 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.9.5

Factor $- 1$ out of $- (2 x^{2}) - 1 (1)$ .

$u = \frac{- (2 x^{2} + 1) - \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.9.6

Factor $- 1$ out of $- \sqrt{8 x^{2} + 1}$ .

$u = \frac{- (2 x^{2} + 1) - (\sqrt{8 x^{2} + 1})}{2 x^{2}}$

Step 2.6.9.7

Factor $- 1$ out of $- (2 x^{2} + 1) - (\sqrt{8 x^{2} + 1})$ .

$u = \frac{- (2 x^{2} + 1 + \sqrt{8 x^{2} + 1})}{2 x^{2}}$

Step 2.6.9.8

Rewrite $- (2 x^{2} + 1 + \sqrt{8 x^{2} + 1})$ as $- 1 (2 x^{2} + 1 + \sqrt{8 x^{2} + 1})$ .

$u = \frac{- 1 (2 x^{2} + 1 + \sqrt{8 x^{2} + 1})}{2 x^{2}}$

Step 2.6.9.9

Move the negative in front of the fraction.

$u = - \frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.10

The final answer is the combination of both solutions.

$u = - \frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$

$u = - \frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.11

Substitute $\cos (6 y)$ for $u$ .

$\cos (6 y) = - \frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}, - \frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.12

Set up each of the solutions to solve for $y$ .

$\cos (6 y) = - \frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$

$\cos (6 y) = - \frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$

Step 2.6.13

Solve for $y$ in $\cos (6 y) = - \frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$ .

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Step 2.6.13.1

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$6 y = \arccos (- \frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.13.2

Simplify the right side.

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Step 2.6.13.2.1

Simplify $\arccos (- \frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}})$ .

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Step 2.6.13.2.1.1

Split the fraction $\frac{2 x^{2} + 1 - \sqrt{8 x^{2} + 1}}{2 x^{2}}$ into two fractions.

$6 y = \arccos (- (\frac{2 x^{2} + 1}{2 x^{2}} + \frac{- \sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.13.2.1.2

Simplify each term.

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Step 2.6.13.2.1.2.1

Split the fraction $\frac{2 x^{2} + 1}{2 x^{2}}$ into two fractions.

$6 y = \arccos (- (\frac{2 x^{2}}{2 x^{2}} + \frac{1}{2 x^{2}} + \frac{- \sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.13.2.1.2.2

Simplify each term.

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Step 2.6.13.2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.6.13.2.1.2.2.1.1

Cancel the common factor.

$6 y = \arccos (- (\frac{2 x^{2}}{2 x^{2}} + \frac{1}{2 x^{2}} + \frac{- \sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.13.2.1.2.2.1.2

Rewrite the expression.

$6 y = \arccos (- (\frac{x^{2}}{x^{2}} + \frac{1}{2 x^{2}} + \frac{- \sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.13.2.1.2.2.2

Cancel the common factor of $x^{2}$ .

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Step 2.6.13.2.1.2.2.2.1

Cancel the common factor.

$6 y = \arccos (- (\frac{x^{2}}{x^{2}} + \frac{1}{2 x^{2}} + \frac{- \sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.13.2.1.2.2.2.2

Rewrite the expression.

$6 y = \arccos (- (1 + \frac{1}{2 x^{2}} + \frac{- \sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.13.2.1.2.3

Move the negative in front of the fraction.

$6 y = \arccos (- (1 + \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.13.2.1.3

Apply the distributive property.

$6 y = \arccos (- 1 \cdot 1 - \frac{1}{2 x^{2}} - - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.13.2.1.4

Simplify.

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Step 2.6.13.2.1.4.1

Multiply $- 1$ by $1$ .

$6 y = \arccos (- 1 - \frac{1}{2 x^{2}} - - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.13.2.1.4.2

Multiply $- - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}$ .

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Step 2.6.13.2.1.4.2.1

Multiply $- 1$ by $- 1$ .

$6 y = \arccos (- 1 - \frac{1}{2 x^{2}} + 1 \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.13.2.1.4.2.2

Multiply $\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}$ by $1$ .

$6 y = \arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.13.3

Divide each term in $6 y = \arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$ by $6$ and simplify.

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Step 2.6.13.3.1

Divide each term in $6 y = \arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$ by $6$ .

$\frac{6 y}{6} = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 2.6.13.3.2

Simplify the left side.

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Step 2.6.13.3.2.1

Cancel the common factor of $6$ .

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Step 2.6.13.3.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 2.6.13.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 2.6.14

Solve for $y$ in $\cos (6 y) = - \frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$ .

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Step 2.6.14.1

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$6 y = \arccos (- \frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.14.2

Simplify the right side.

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Step 2.6.14.2.1

Simplify $\arccos (- \frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}})$ .

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Step 2.6.14.2.1.1

Split the fraction $\frac{2 x^{2} + 1 + \sqrt{8 x^{2} + 1}}{2 x^{2}}$ into two fractions.

$6 y = \arccos (- (\frac{2 x^{2} + 1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.14.2.1.2

Simplify each term.

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Step 2.6.14.2.1.2.1

Split the fraction $\frac{2 x^{2} + 1}{2 x^{2}}$ into two fractions.

$6 y = \arccos (- (\frac{2 x^{2}}{2 x^{2}} + \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.14.2.1.2.2

Simplify each term.

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Step 2.6.14.2.1.2.2.1

Cancel the common factor of $2$ .

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Step 2.6.14.2.1.2.2.1.1

Cancel the common factor.

$6 y = \arccos (- (\frac{2 x^{2}}{2 x^{2}} + \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.14.2.1.2.2.1.2

Rewrite the expression.

$6 y = \arccos (- (\frac{x^{2}}{x^{2}} + \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.14.2.1.2.2.2

Cancel the common factor of $x^{2}$ .

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Step 2.6.14.2.1.2.2.2.1

Cancel the common factor.

$6 y = \arccos (- (\frac{x^{2}}{x^{2}} + \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.14.2.1.2.2.2.2

Rewrite the expression.

$6 y = \arccos (- (1 + \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}}))$

Step 2.6.14.2.1.3

Simplify by multiplying through.

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Step 2.6.14.2.1.3.1

Apply the distributive property.

$6 y = \arccos (- 1 \cdot 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.14.2.1.3.2

Multiply $- 1$ by $1$ .

$6 y = \arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$

Step 2.6.14.3

Divide each term in $6 y = \arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$ by $6$ and simplify.

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Step 2.6.14.3.1

Divide each term in $6 y = \arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$ by $6$ .

$\frac{6 y}{6} = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 2.6.14.3.2

Simplify the left side.

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Step 2.6.14.3.2.1

Cancel the common factor of $6$ .

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Step 2.6.14.3.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 2.6.14.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 2.6.15

List all of the solutions.

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

$y = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}, \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$

Step 4

Verify if $f^{- 1} (x) = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}, \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$ is the inverse of $f (x) = - \frac{\sqrt{1 - \cos (6 x)}}{1 + \cos (6 x)}$ .

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Step 4.1

The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of $f (x) = - \frac{\sqrt{1 - \cos (6 x)}}{1 + \cos (6 x)}$ and $f^{- 1} (x) = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}, \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$ and compare them.

Step 4.2

Find the domain of $\frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$ .

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Step 4.2.1

Set the radicand in $\sqrt{8 x^{2} + 1}$ greater than or equal to $0$ to find where the expression is defined.

$8 x^{2} + 1 \geq 0$

Step 4.2.2

Solve for $x$ .

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Step 4.2.2.1

Subtract $1$ from both sides of the inequality.

$8 x^{2} \geq - 1$

Step 4.2.2.2

Divide each term in $8 x^{2} \geq - 1$ by $8$ and simplify.

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Step 4.2.2.2.1

Divide each term in $8 x^{2} \geq - 1$ by $8$ .

$\frac{8 x^{2}}{8} \geq \frac{- 1}{8}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $8$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{8 x^{2}}{8} \geq \frac{- 1}{8}$

Step 4.2.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} \geq \frac{- 1}{8}$

Step 4.2.2.2.3

Simplify the right side.

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Step 4.2.2.2.3.1

Move the negative in front of the fraction.

$x^{2} \geq - \frac{1}{8}$

Step 4.2.2.3

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 4.2.3

Set the argument in $\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$ greater than or equal to $- 1$ to find where the expression is defined.

$- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \geq - 1$

Step 4.2.4

Solve for $x$ .

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Step 4.2.4.1

Solve for $\sqrt{8 x^{2} + 1}$ .

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Step 4.2.4.1.1

Move all terms not containing $\sqrt{8 x^{2} + 1}$ to the right side of the inequality.

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Step 4.2.4.1.1.1

Add $1$ to both sides of the inequality.

$- \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \geq - 1 + 1$

Step 4.2.4.1.1.2

Add $\frac{1}{2 x^{2}}$ to both sides of the inequality.

$\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \geq - 1 + 1 + \frac{1}{2 x^{2}}$

Step 4.2.4.1.1.3

Add $- 1$ and $1$ .

$\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \geq 0 + \frac{1}{2 x^{2}}$

Step 4.2.4.1.1.4

Add $0$ and $\frac{1}{2 x^{2}}$ .

$\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \geq \frac{1}{2 x^{2}}$

Step 4.2.4.1.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sqrt{8 x^{2} + 1} \geq 1$

Step 4.2.4.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{8 x^{2} + 1}}^{2} \geq 1^{2}$

Step 4.2.4.3

Simplify each side of the inequality.

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Step 4.2.4.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x^{2} + 1}$ as ${(8 x^{2} + 1)}^{\frac{1}{2}}$ .

${({(8 x^{2} + 1)}^{\frac{1}{2}})}^{2} \geq 1^{2}$

Step 4.2.4.3.2

Simplify the left side.

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Step 4.2.4.3.2.1

Simplify ${({(8 x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.4.3.2.1.1

Multiply the exponents in ${({(8 x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.4.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(8 x^{2} + 1)}^{\frac{1}{2} \cdot 2} \geq 1^{2}$

Step 4.2.4.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.4.3.2.1.1.2.1

Cancel the common factor.

${(8 x^{2} + 1)}^{\frac{1}{2} \cdot 2} \geq 1^{2}$

Step 4.2.4.3.2.1.1.2.2

Rewrite the expression.

${(8 x^{2} + 1)}^{1} \geq 1^{2}$

Step 4.2.4.3.2.1.2

Simplify.

$8 x^{2} + 1 \geq 1^{2}$

Step 4.2.4.3.3

Simplify the right side.

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Step 4.2.4.3.3.1

One to any power is one.

$8 x^{2} + 1 \geq 1$

Step 4.2.4.4

Solve for $x$ .

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Step 4.2.4.4.1

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.2.4.4.1.1

Subtract $1$ from both sides of the inequality.

$8 x^{2} \geq 1 - 1$

Step 4.2.4.4.1.2

Subtract $1$ from $1$ .

$8 x^{2} \geq 0$

Step 4.2.4.4.2

Divide each term in $8 x^{2} \geq 0$ by $8$ and simplify.

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Step 4.2.4.4.2.1

Divide each term in $8 x^{2} \geq 0$ by $8$ .

$\frac{8 x^{2}}{8} \geq \frac{0}{8}$

Step 4.2.4.4.2.2

Simplify the left side.

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Step 4.2.4.4.2.2.1

Cancel the common factor of $8$ .

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Step 4.2.4.4.2.2.1.1

Cancel the common factor.

$\frac{8 x^{2}}{8} \geq \frac{0}{8}$

Step 4.2.4.4.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} \geq \frac{0}{8}$

Step 4.2.4.4.2.3

Simplify the right side.

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Step 4.2.4.4.2.3.1

Divide $0$ by $8$ .

$x^{2} \geq 0$

Step 4.2.4.4.3

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 4.2.5

Set the argument in $\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})$ less than or equal to $1$ to find where the expression is defined.

$- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \leq 1$

Step 4.2.6

Solve for $x$ .

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Step 4.2.6.1

Solve for $\sqrt{8 x^{2} + 1}$ .

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Step 4.2.6.1.1

Move all terms not containing $\sqrt{8 x^{2} + 1}$ to the right side of the inequality.

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Step 4.2.6.1.1.1

Add $1$ to both sides of the inequality.

$- \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \leq 1 + 1$

Step 4.2.6.1.1.2

Add $\frac{1}{2 x^{2}}$ to both sides of the inequality.

$\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \leq 1 + 1 + \frac{1}{2 x^{2}}$

Step 4.2.6.1.1.3

Add $1$ and $1$ .

$\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} \leq 2 + \frac{1}{2 x^{2}}$

Step 4.2.6.1.2

Multiply both sides by $2 x^{2}$ .

$\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} (2 x^{2}) \leq (2 + \frac{1}{2 x^{2}}) (2 x^{2})$

Step 4.2.6.1.3

Simplify.

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Step 4.2.6.1.3.1

Simplify the left side.

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Step 4.2.6.1.3.1.1

Simplify $\frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} (2 x^{2})$ .

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Step 4.2.6.1.3.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} x^{2} \leq (2 + \frac{1}{2 x^{2}}) (2 x^{2})$

Step 4.2.6.1.3.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.6.1.3.1.1.2.1

Factor $2$ out of $2 x^{2}$ .

$2 \frac{\sqrt{8 x^{2} + 1}}{2 (x^{2})} x^{2} \leq (2 + \frac{1}{2 x^{2}}) (2 x^{2})$

Step 4.2.6.1.3.1.1.2.2

Cancel the common factor.

$2 \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}} x^{2} \leq (2 + \frac{1}{2 x^{2}}) (2 x^{2})$

Step 4.2.6.1.3.1.1.2.3

Rewrite the expression.

$\frac{\sqrt{8 x^{2} + 1}}{x^{2}} x^{2} \leq (2 + \frac{1}{2 x^{2}}) (2 x^{2})$

Step 4.2.6.1.3.1.1.3

Cancel the common factor of $x^{2}$ .

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Step 4.2.6.1.3.1.1.3.1

Cancel the common factor.

$\frac{\sqrt{8 x^{2} + 1}}{x^{2}} x^{2} \leq (2 + \frac{1}{2 x^{2}}) (2 x^{2})$

Step 4.2.6.1.3.1.1.3.2

Rewrite the expression.

$\sqrt{8 x^{2} + 1} \leq (2 + \frac{1}{2 x^{2}}) (2 x^{2})$

Step 4.2.6.1.3.2

Simplify the right side.

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Step 4.2.6.1.3.2.1

Simplify $(2 + \frac{1}{2 x^{2}}) (2 x^{2})$ .

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Step 4.2.6.1.3.2.1.1

Simplify by multiplying through.

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Step 4.2.6.1.3.2.1.1.1

Apply the distributive property.

$\sqrt{8 x^{2} + 1} \leq 2 (2 x^{2}) + \frac{1}{2 x^{2}} (2 x^{2})$

Step 4.2.6.1.3.2.1.1.2

Simplify the expression.

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Step 4.2.6.1.3.2.1.1.2.1

Multiply $2$ by $2$ .

$\sqrt{8 x^{2} + 1} \leq 4 x^{2} + \frac{1}{2 x^{2}} (2 x^{2})$

Step 4.2.6.1.3.2.1.1.2.2

Rewrite using the commutative property of multiplication.

$\sqrt{8 x^{2} + 1} \leq 4 x^{2} + 2 \frac{1}{2 x^{2}} x^{2}$

Step 4.2.6.1.3.2.1.2

Simplify each term.

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Step 4.2.6.1.3.2.1.2.1

Cancel the common factor of $2$ .

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Step 4.2.6.1.3.2.1.2.1.1

Factor $2$ out of $2 x^{2}$ .

$\sqrt{8 x^{2} + 1} \leq 4 x^{2} + 2 \frac{1}{2 (x^{2})} x^{2}$

Step 4.2.6.1.3.2.1.2.1.2

Cancel the common factor.

$\sqrt{8 x^{2} + 1} \leq 4 x^{2} + 2 \frac{1}{2 x^{2}} x^{2}$

Step 4.2.6.1.3.2.1.2.1.3

Rewrite the expression.

$\sqrt{8 x^{2} + 1} \leq 4 x^{2} + \frac{1}{x^{2}} x^{2}$

Step 4.2.6.1.3.2.1.2.2

Cancel the common factor of $x^{2}$ .

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Step 4.2.6.1.3.2.1.2.2.1

Cancel the common factor.

$\sqrt{8 x^{2} + 1} \leq 4 x^{2} + \frac{1}{x^{2}} x^{2}$

Step 4.2.6.1.3.2.1.2.2.2

Rewrite the expression.

$\sqrt{8 x^{2} + 1} \leq 4 x^{2} + 1$

Step 4.2.6.2

To remove the radical on the left side of the inequality, square both sides of the inequality.

${\sqrt{8 x^{2} + 1}}^{2} \leq {(4 x^{2} + 1)}^{2}$

Step 4.2.6.3

Simplify each side of the inequality.

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Step 4.2.6.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x^{2} + 1}$ as ${(8 x^{2} + 1)}^{\frac{1}{2}}$ .

${({(8 x^{2} + 1)}^{\frac{1}{2}})}^{2} \leq {(4 x^{2} + 1)}^{2}$

Step 4.2.6.3.2

Simplify the left side.

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Step 4.2.6.3.2.1

Simplify ${({(8 x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.6.3.2.1.1

Multiply the exponents in ${({(8 x^{2} + 1)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.6.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(8 x^{2} + 1)}^{\frac{1}{2} \cdot 2} \leq {(4 x^{2} + 1)}^{2}$

Step 4.2.6.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.6.3.2.1.1.2.1

Cancel the common factor.

${(8 x^{2} + 1)}^{\frac{1}{2} \cdot 2} \leq {(4 x^{2} + 1)}^{2}$

Step 4.2.6.3.2.1.1.2.2

Rewrite the expression.

${(8 x^{2} + 1)}^{1} \leq {(4 x^{2} + 1)}^{2}$

Step 4.2.6.3.2.1.2

Simplify.

$8 x^{2} + 1 \leq {(4 x^{2} + 1)}^{2}$

Step 4.2.6.3.3

Simplify the right side.

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Step 4.2.6.3.3.1

Simplify ${(4 x^{2} + 1)}^{2}$ .

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Step 4.2.6.3.3.1.1

Rewrite ${(4 x^{2} + 1)}^{2}$ as $(4 x^{2} + 1) (4 x^{2} + 1)$ .

$8 x^{2} + 1 \leq (4 x^{2} + 1) (4 x^{2} + 1)$

Step 4.2.6.3.3.1.2

Expand $(4 x^{2} + 1) (4 x^{2} + 1)$ using the FOIL Method.

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Step 4.2.6.3.3.1.2.1

Apply the distributive property.

$8 x^{2} + 1 \leq 4 x^{2} (4 x^{2} + 1) + 1 (4 x^{2} + 1)$

Step 4.2.6.3.3.1.2.2

Apply the distributive property.

$8 x^{2} + 1 \leq 4 x^{2} (4 x^{2}) + 4 x^{2} \cdot 1 + 1 (4 x^{2} + 1)$

Step 4.2.6.3.3.1.2.3

Apply the distributive property.

$8 x^{2} + 1 \leq 4 x^{2} (4 x^{2}) + 4 x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1$

Step 4.2.6.3.3.1.3

Simplify and combine like terms.

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Step 4.2.6.3.3.1.3.1

Simplify each term.

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Step 4.2.6.3.3.1.3.1.1

Rewrite using the commutative property of multiplication.

$8 x^{2} + 1 \leq 4 \cdot 4 x^{2} x^{2} + 4 x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1$

Step 4.2.6.3.3.1.3.1.2

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 4.2.6.3.3.1.3.1.2.1

Move $x^{2}$ .

$8 x^{2} + 1 \leq 4 \cdot 4 (x^{2} x^{2}) + 4 x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1$

Step 4.2.6.3.3.1.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$8 x^{2} + 1 \leq 4 \cdot 4 x^{2 + 2} + 4 x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1$

Step 4.2.6.3.3.1.3.1.2.3

Add $2$ and $2$ .

$8 x^{2} + 1 \leq 4 \cdot 4 x^{4} + 4 x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1$

Step 4.2.6.3.3.1.3.1.3

Multiply $4$ by $4$ .

$8 x^{2} + 1 \leq 16 x^{4} + 4 x^{2} \cdot 1 + 1 (4 x^{2}) + 1 \cdot 1$

Step 4.2.6.3.3.1.3.1.4

Multiply $4$ by $1$ .

$8 x^{2} + 1 \leq 16 x^{4} + 4 x^{2} + 1 (4 x^{2}) + 1 \cdot 1$

Step 4.2.6.3.3.1.3.1.5

Multiply $4 x^{2}$ by $1$ .

$8 x^{2} + 1 \leq 16 x^{4} + 4 x^{2} + 4 x^{2} + 1 \cdot 1$

Step 4.2.6.3.3.1.3.1.6

Multiply $1$ by $1$ .

$8 x^{2} + 1 \leq 16 x^{4} + 4 x^{2} + 4 x^{2} + 1$

Step 4.2.6.3.3.1.3.2

Add $4 x^{2}$ and $4 x^{2}$ .

$8 x^{2} + 1 \leq 16 x^{4} + 8 x^{2} + 1$

Step 4.2.6.4

Solve for $x$ .

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Step 4.2.6.4.1

Rewrite so $x$ is on the left side of the inequality.

$16 x^{4} + 8 x^{2} + 1 \geq 8 x^{2} + 1$

Step 4.2.6.4.2

Move all terms containing $x$ to the left side of the inequality.

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Step 4.2.6.4.2.1

Subtract $8 x^{2}$ from both sides of the inequality.

$16 x^{4} + 8 x^{2} + 1 - 8 x^{2} \geq 1$

Step 4.2.6.4.2.2

Combine the opposite terms in $16 x^{4} + 8 x^{2} + 1 - 8 x^{2}$ .

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Step 4.2.6.4.2.2.1

Subtract $8 x^{2}$ from $8 x^{2}$ .

$16 x^{4} + 0 + 1 \geq 1$

Step 4.2.6.4.2.2.2

Add $16 x^{4}$ and $0$ .

$16 x^{4} + 1 \geq 1$

Step 4.2.6.4.3

Move all terms not containing $x$ to the right side of the inequality.

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Step 4.2.6.4.3.1

Subtract $1$ from both sides of the inequality.

$16 x^{4} \geq 1 - 1$

Step 4.2.6.4.3.2

Subtract $1$ from $1$ .

$16 x^{4} \geq 0$

Step 4.2.6.4.4

Divide each term in $16 x^{4} \geq 0$ by $16$ and simplify.

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Step 4.2.6.4.4.1

Divide each term in $16 x^{4} \geq 0$ by $16$ .

$\frac{16 x^{4}}{16} \geq \frac{0}{16}$

Step 4.2.6.4.4.2

Simplify the left side.

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Step 4.2.6.4.4.2.1

Cancel the common factor of $16$ .

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Step 4.2.6.4.4.2.1.1

Cancel the common factor.

$\frac{16 x^{4}}{16} \geq \frac{0}{16}$

Step 4.2.6.4.4.2.1.2

Divide $x^{4}$ by $1$ .

$x^{4} \geq \frac{0}{16}$

Step 4.2.6.4.4.3

Simplify the right side.

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Step 4.2.6.4.4.3.1

Divide $0$ by $16$ .

$x^{4} \geq 0$

Step 4.2.6.4.5

Since the left side has an even power, it is always positive for all real numbers.

All real numbers

Step 4.2.7

Set the denominator in $\frac{1}{2 x^{2}}$ equal to $0$ to find where the expression is undefined.

$2 x^{2} = 0$

Step 4.2.8

Solve for $x$ .

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Step 4.2.8.1

Divide each term in $2 x^{2} = 0$ by $2$ and simplify.

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Step 4.2.8.1.1

Divide each term in $2 x^{2} = 0$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Step 4.2.8.1.2

Simplify the left side.

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Step 4.2.8.1.2.1

Cancel the common factor of $2$ .

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Step 4.2.8.1.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{0}{2}$

Step 4.2.8.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{2}$

Step 4.2.8.1.3

Simplify the right side.

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Step 4.2.8.1.3.1

Divide $0$ by $2$ .

$x^{2} = 0$

Step 4.2.8.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 4.2.8.3

Simplify $\pm \sqrt{0}$ .

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Step 4.2.8.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 4.2.8.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 4.2.8.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 4.2.9

The domain is all values of $x$ that make the expression defined.

$(- \infty, 0) \cup (0, \infty)$

Step 4.3

Since the domain of $f^{- 1} (x) = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}, \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$ is not equal to the range of $f (x) = - \frac{\sqrt{1 - \cos (6 x)}}{1 + \cos (6 x)}$ , then $f^{- 1} (x) = \frac{\arccos (- 1 - \frac{1}{2 x^{2}} + \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}, \frac{\arccos (- 1 - \frac{1}{2 x^{2}} - \frac{\sqrt{8 x^{2} + 1}}{2 x^{2}})}{6}$ is not an inverse of $f (x) = - \frac{\sqrt{1 - \cos (6 x)}}{1 + \cos (6 x)}$ .

There is no inverse

Step 5